Basic Machine Theory

The generalized machine model transforms the stator windings into equivalent commutator windings, using the dq0 transformation as follows:

The three-phase rotor winding may also be transformed into a two-phase equivalent winding, with additional windings added to each axis to fully represent that particular machine, as is shown in Figure 8-1. For more details on these transformations, please see [12] and [13].

Figure 7-1 - Conceptual Diagram of the Three-Phase and dq Windings

NOTE:  All quantities shown in Figure 7-1 are in pu

Where,

Support subroutines are included in the machine model library for calculating the equivalent circuit parameters of a synchronous machine from commonly supplied data. Typical parameters are supplied for small, medium and large squirrel cage motors.

The d-axis equivalent circuit for the generalized machine is shown in Figure 7-2. Figure 7-3 illustrates the flux paths associated with various d-axis inductances:

Figure 7-2 - d-axis Equivalent Circuit

Figure 7-3 - Flux Paths Associated with Various d-axis Inductances

Referring to Figures 7-2 and 7-3, the following equations can be derived:

Where,

Similar equations hold for the q-axis except the speed voltage term, , is positive, and:

Inversion of Equation 7-2 gives the standard state variable form with state vector X consisting of the currents, and the input vector U, applied voltages.  That is:

In the above form, Equations 7-7 and 7-8 are particularly easy to integrate.  The equations are solved using trapezoidal integration to obtain the currents.  The torque equation is given as:

and the mechanical dynamic equation for motor operation is: